A fallacious proof that 0 = 1

This is an old one but perhaps you haven't seen it?

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1. Start with the equation:                    x = y 
2. Multiply both sides by x:                  x² = xy
3. Subtract y² from both sides:      x² - y² = xy - y²
4. Factor both sides:           (x - y)(x + y) = y(x - y)
5. Divide both sides by (x - y):        x + y = y
6. Since x = y, substitute y for x:     y + y = y
7. Simplify:                                             2y = y
8. Divide both sides by y:                      2 = 1
9. Subtract 1 from both sides:              1 = 0

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I asked ChatGPT for some better fallacies but in vain. It just produced stupid stuff.

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But then there is this - integration by parts:

∫ (dx/x) = ∫ (1) (1/x) dx

= x (1/x) - ∫ x (-1/x²) dx

= 1 + ∫ (dx/x)

⟹ 0 = 1.

Two appearances of the same integral can (and should!) have (different) constants of integration; or, as someone said: "From C to shining C".

Try it 'definitely' and you will see it comes out right.

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